mvee_REX: Minimum-volume enclosing ellipsoid

In OptimalDesign: A Toolbox for Computing Efficient Designs of Experiments

Description

Computes the shape matrix H and the center z of the minimum-volume ellipsoid enclosing a finite set of data-points.

Usage

mvee_REX(Data, alg.AA="REX", eff=0.999999, it.max=Inf, t.max=60,
         picture=FALSE, echo=TRUE, track=TRUE)

Arguments

Data	the n times d (where d<n) matrix containing the d-dimensional data-vectors as rows.
alg.AA	the underlying computational method for approximate D-optimal design; possible values are "REX", "MUL" and "VDM".
eff	the minimum required efficiency.
it.max	a limit on the number of iterations of the underlying D-optimum approximate design algorithm.
t.max	a limit on the time of computation.
picture	Should a picture be plotted? (For the picture, the data need to be either two- or three-dimensional.)
echo	Print the call of the function?
track	Display the progress of the computation?

Details

The problem of the minimum-volume data-enclosing ellipsoid (MVEE) is computationally equivalent to the problem of D-optimal approximate design for an artificial problem based on the data. This procedure performs the computation and the proper conversion of the D-optimal approximate design to the MVEE parameters (the center and the shape matrix).

Value

Output is a list with components:

call	the call of the function
H	the shape matrix of the MVEE
z	the center of the MVEE
bpts	a set containing the boundary points of the MVEE
vol	the volume of the MVEE
eff.best	the actual precision of the result (1 is the perfect precision)
t.iter	the number of iterations of the underlying D-optimum design algorithm
t.act	the actual time of the computation

Note

Note: The affine hull of the rows of X should be the full space of dimension d. For the choice of the algorithm, see the comments in od_REX.

Author(s)

Radoslav Harman, Lenka Filova

References

Harman R, Filova L, Richtarik P (2019). A randomized exchange algorithm for computing optimal approximate designs of experiments. Journal of the American Statistical Association, 1-30.

See Also

od_REX

Examples

# Generate random 1000 points in a 3-dimensional space
# and compute the MVEE

Data <- matrix(rnorm(3000), ncol = 3)
mvee_REX(Data, picture = FALSE)

Example output

[1] Call of the function:
mvee_REX(Data = Data, picture = FALSE)
[1] Call of the function:
od_REX(Fx = Fx, eff = eff, it.max = it.max, t.max = t.max, track = track)
[1] Running od_D_REX for cca 60 seconds starting at 2021-04-03 08:57:32.
[1] The problem size is n=1000 and m=4.
[1] od_D_REX Time: 0.01 Efficiency: 0.761630771
[1] od_D_REX Time: 0.01 Efficiency: 0.801047391
[1] od_D_REX Time: 0.02 Efficiency: 0.951061779
[1] od_D_REX Time: 0.02 Efficiency: 0.99199223
[1] od_D_REX Time: 0.03 Efficiency: 0.998249893
[1] od_D_REX Time: 0.04 Efficiency: 0.99988339
[1] od_D_REX Time: 0.05 Efficiency: 0.999996362
[1] od_D_REX Time: 0.05 Efficiency: 0.999999879
[1] od_D_REX finished at 2021-04-03 08:57:33
[1] Computation time: 0.05
[1] D-criterion value: 2.99350330826406
[1] Efficiency at least: 0.999999879294824
[1] Call of the function:
od_DEL(Fx = Fx, w = res$w.best)
$call
mvee_REX(Data = Data, picture = FALSE)

$H
             [,1]         [,2]        [,3]
[1,]  0.072748757 -0.007137668 0.025471499
[2,] -0.007137668  0.089074005 0.008608322
[3,]  0.025471499  0.008608322 0.082055590

$z
[1]  0.08254455 -0.08369861  0.01873410

$bpts
[1] 106 122 159 181 352 496 548 866

$vol
[1] 195.0428

$eff.best
[1] 0.9999999

$n.iter
[1] 8

$t.act
[1] 0.05

OptimalDesign documentation built on March 26, 2020, 9:35 p.m.